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Phase detector

The goal of the PLL is to maintain a demodulating sine and cosine that match the incoming carrier. Suppose ω c is the believed digital carrier frequency. We can then represent the actual received carrier frequency as theexpected carrier frequency with some offset, ω c ˜ ω c θ ˜ n . The NCO generates the demodulating sine and cosine with the expected digital frequency ω c and offsets this frequency with the output of the loop filter. The NCO frequency can then be modeled as ω c ^ ω c θ ^ n . Using the appropriate trigonometric identities A B 1 2 A B A B and A B 1 2 B A A B . , the in-phase and quadrature signals can be expressed as

z 0 n 1 2 θ ˜ n θ ^ n 2 ω c θ ˜ n θ ^ n
z Q n 1 2 θ ˜ n θ ^ n 2 ω c θ ˜ n θ ^ n
After applying a low-pass filter to remove the double frequency terms, we have
y 1 n 1 2 θ ˜ n θ ^ n
y Q n 1 2 θ ˜ n θ ^ n
Note that the quadrature signal, z Q n , is zero when the received carrier and internallygenerated waves are exactly matched in frequency and phase. When the phases are only slightly mismatched we can use therelation
θ small θ θ
and let the current value of the quadrature channel approximate the phase difference: z Q n θ ˜ n θ ^ n . With the exception of the sign error, this difference is essentially how much we need to offset our NCOfrequency If θ ˜ n θ ^ n 0 then θ ^ n is too large and we want to decrease our NCO phase. . To make sure that the sign of the phase estimate is right, in this example the phase detector issimply negative one times the value of the quadrature signal. In a more advanced receiver, information from boththe in-phase and quadrature branches is used to generate an estimate of the phase error. What should the relationship between the I and Q branches be fora digital QPSK signal?

Loop filter

The estimated phase mismatch estimate is fed to the NCO via a loop filter, often a simple low-pass filter. For thisexercise you can use a one-tap IIR filter,

y n β x n α y n 1
To ensure unity gain at DC, we select β 1 α

It is suggested that you start by choosing α 0.6 and K 0.15 for the loop gain. Once you have a working system, investigate the effects of modifying these values.

Matlab simulation

Simulate the PLL system shown in [link] using MATLAB. As with the DLL simulation, you will have to simulate the PLL on a sample-by-sample basis.

Use [link] to implement your NCO in MATLAB. However, to ensure that the phase term does not grow toinfinity, you should use addition modulo 2 in the phase update relation. This can be done by setting θ n θ n 2 whenever θ n 2 .

[link] illustrates how the proposed PLL will behave when given a modulated BPSK waveform. In this case thetransmitted carrier frequency was set to ω c ˜ 2 1024 to simulate a frequency offset.

Output of PLL sub-system for BPSK modulated carrier.

Note that an amplitude transition in the BPSK waveform is equivalent to a phase shift of the carrier by 2 . Immediately after this phase change occurs, the PLL begins to adjust the phase to force the quadraturecomponent to zero (and the in-phase component to 1 2 ). Why would this phase detector not work in a real BPSK environment? How could it be changed to work?

Questions & Answers

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Source:  OpenStax, Digital signal processing laboratory (ece 420). OpenStax CNX. Sep 27, 2006 Download for free at http://cnx.org/content/col10236/1.14
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